Rotating Black Holes in Higher Order Gravity

TL;DR

This paper introduces a new method for constructing rotating black hole solutions with stealth scalar hair in higher order gravity theories, revealing novel scalar-dressed black holes with finite scalar fields at horizons.

Contribution

It develops a technique for finding black hole solutions with stealth hair and presents a new rotating black hole solution with finite scalar fields at horizons in scalar-tensor theories.

Findings

Existence of Einstein metrics with stealth hair under specific geodesic conditions

Construction of a scalar-dressed rotating black hole with finite scalar field at horizons

New insights into black hole solutions in higher order gravity theories

Abstract

We develop a new technique for finding black hole solutions in modified gravity that have "stealth" hair, i.e., hair whose only gravitational effect is to tune the cosmological constant. We consider scalar-tensor theories in which gravitational waves propagate at the speed of light, and show that Einstein metrics can be painted with stealth hair provided there exists a family of geodesics always normal to spacelike surfaces. We also present a novel scalar-dressed rotating black hole that has finite scalar field at both the black hole and cosmological event horizons.

Equations

$$\mathcal{L}$$

$$+A_4{\varphi }^{\mu }{\varphi }_{\mu \rho }{\varphi }^{\rho \nu }{\varphi }_{\nu }+A_5({\varphi }^{\mu }{\varphi }_{\mu \nu }{\varphi }^{\nu }{)}^2,$$

$$A_4$$

$$A_5$$

$$\left[A_3(X_0)\left({\mathcal{E}}_3-\Lambda X_0\right)-2\left(K_X+4\Lambda G_X\right){|}_{{}_{X_0}}\right]{\varphi }_{\mu }{\varphi }_{\nu }$$

$$+\left(K+2\Lambda G\right){|}_{{}_{X_0}}{g}_{\mu \nu }=0$$

$$A_3(X_0)\left({\mathcal{E}}_4+2R_{\mu \nu \rho \sigma }{\varphi }^{\nu \sigma }{\varphi }^{\mu }{\varphi }^{\rho }-3\Lambda X_0\square \varphi \right)$$

$$-2\left(K_X+4\Lambda G_X\right){|}_{{}_{X_0}}\square \varphi =0$$

$${\mathcal{E}}_3$$

$${\mathcal{E}}_4$$

$$(K_X+4\Lambda G_X){|}_{X_0}=0,$$

$$\frac{d^2{x}^{\mu }}{d{\lambda }^2}+{\Gamma }_{\rho \sigma }^{\mu }\frac{d{x}^{\rho }}{d\lambda }\frac{d{x}^{\sigma }}{d\lambda }=0$$

$$\frac{\partial S}{\partial x^{\mu }}=p_{\mu }=g_{\mu \nu }\frac{d{x}^{\nu }}{d\lambda }.$$

$$\varphi \leftrightarrow S,$$

$$d{s}^2$$

$$+\frac{{\Delta }_{\theta }{\text{sin}}^2\theta }{{\Xi }^2{\rho }^2}{\left[adt-\left(r^2+a^2\right)d\phi \right]}^2,$$

$${\Delta }_r$$

$${\Delta }_{\theta }$$

$$\mathcal{Q}$$

$$-{\Xi }^2\left[{\left(aE-L_z\right)}^2-\frac{{\text{sin}}^2\theta }{{\Delta }_{\theta }}{\left(aE-\frac{L_z}{{\text{sin}}^2\theta }\right)}^2\right],$$

$$S=-Et+L_z\phi +S_r(r)+S_{\theta }(\theta ),$$

$$S_r=\pm \int \frac{\sqrt{R}}{{\Delta }_r}dr,S_{\theta }=\pm \int \frac{\sqrt{\Theta }}{{\Delta }_{\theta }}d\theta ,$$

$$R$$

$$\Theta$$

$$R$$

$$S_r=(H[r-r_0]-H[r_0-r]){\int }_{r_0}^r\frac{|\sqrt{R}|dr}{{\Delta }_r}$$

$$\pm S_{\theta }=$$

$$-\text{log}\left[\frac{\sqrt{1-{\eta }^2+\frac{a^2}{{\ell }^2}{\text{cos}}^2\theta }+\frac{a}{\ell }\text{cos}\theta }{\sqrt{1-{\eta }^2}}\right],$$

$$S_{\theta }=\pm \frac{m\ell }{2}\text{log}{\Delta }_{\theta },$$

$$\varphi =-m\left[{\eta }_{c}t+\int \frac{r(r-r_0)\sqrt{r(r+2r_0)}}{\ell {\Delta }_r}dr\right],$$

$$(K_X+4\Lambda G_X+\frac{3}{2}\Lambda X{A}_3){|}_{X_0}=0,$$

$${\varphi }_s=S_r(r)=\sqrt{X_0}\int \frac{r}{\sqrt{{\Delta }_r}}dr,$$

Full text

Black Holes are a cornerstone in the study of General Relativity (GR), be it theoretical or practical. From the theoretical perspective, we have several exact solutions corresponding to black holes in GR, with charges such as mass, angular momentum, and even acceleration or Taub NUT parameters. Of these, the most important is the Kerr solution that describes the rotating black hole; most astrophysical black holes are believed to be rotating, indeed, the first detection of gravitational waves was from the merger of two spinning Kerr black holes Abbott:2016blz .

Another key objective in gravity is to explain our universe. One of the main challenges in cosmology is to explain late time acceleration: Is it due to dark energy, modified gravity, or a very small cosmological constant $\Lambda$, finely tuned by some as yet undiscovered mechanism? One fruitful means of explaining the small late time acceleration is to modify gravity in such a way as to induce (or tune) a cosmological constant, but if gravity is modified, then it is vital to understand how these modifications affect black holes. If it is not possible to construct astrophysically realistic black hole solutions, then the theory cannot be considered viable.

In GR, the Kerr solution (see Carter:1968ks ; CARTER1968399 for inclusion of a cosmological constant) describes the most general axisymmetric stationary rotating black hole, moreover, the event horizon telescope Doeleman:2009te hopes soon to be able to directly image the shadow of the black hole at the centre of our galaxy, possibly setting constraints on deviations from Kerr Johannsen:2016vqy . It is important therefore to investigate whether spinning black hole solutions exist in modified gravity, and if so, do they carry non-trivial extra degrees of freedom?

In this letter, we focus on a particular family of modified scalar-tensor theories of gravity, developing a new technique to find, for the first time, astrophysically realistic rotating black hole solutions with a nontrivial scalar field, or stealth hair. Our method is based on the Hamilton-Jacobi approach to finding geodesics. In brief, we prove that a GR solution can also be a solution to modified gravity if the scalar field is the Hamilton-Jacobi potential for a geodesic congruence in the spacetime in question. This allows for a wide range of physically interesting stealth solutions, in particular we present examples of rotating black holes with stealth hair.

We focus for definiteness on shift-symmetric scalar-tensor theories of gravity in the family of Degenerate Higher Order Scalar-Tensor (DHOST) theories Langlois:2015cwa ; Crisostomi:2016czh ; BenAchour:2016fzp , although the technique can be generalised to other modifications of gravity. In particular we will focus on the subset of theories where gravitational waves propagate at the speed of light, $c_{T}=1$, in accord with the recent multi-messenger neutron star binary merger observation GBM:2017lvd .

The most general shift-symmetric scalar-tensor theory of gravity in which gravitational waves propagate with the speed of light is

[TABLE]

where $K$, $G$ and $A_{i}$ are all functions of $X=(\partial\phi)^{2}$, and we abbreviate $\partial_{\mu}\phi$ as $\phi_{\mu}$, and $\nabla_{\nu}\nabla_{\mu}\phi$ as $\phi_{\mu\nu}$. In order to propagate a single scalar degree of freedom and avoid Ostrogradski instabilities, $A_{4,5}$ are constrained by

[TABLE]

where $G_{X}=\partial G/\partial X$. These constraints reduce the number of free functions to three ($K,G$ and $A_{3}$) and for simplicity we do not consider the cubic Horndeski term in our Lagrangian (1), though this can easily be reinstated if required.

A family of exact static spherically symmetric solutions, with the scalar field playing the role of dark energy, was initially found in a class of shift-symmetric Horndeski theories Babichev:2013cya . It has the nice feature of locally describing a Schwarzschild geometry while asymptotically approaching a self-tuned accelerating cosmology. Since these solutions acquire a metric similar to that of GR while having a non trivial scalar field, they have been widely called stealth solutions111These solutions were extended and generalised in different modified gravity theories with similar properties, see for example Rinaldi:2012vy ; Kobayashi:2014eva ; Babichev:2017guv ; Chagoya:2016aar ; Babichev:2017rti .. They can be mapped via disformal transformations to stealth solutions of DHOST theories with unitary speed of gravitational waves Babichev:2017lmw , and are free of ghost and gradient instabilities Babichev:2018uiw . For spherically symmetric stealth solutions in DHOST theories see Chagoya:2018lmv ; BenAchour:2018dap ; Motohashi:2019sen .

Known stealth solutions are spherically symmetric and all feature the same characteristic: a constant kinetic term, $X$, for the scalar field that does not deform an underlying Einstein geometry. We suspect that it is this feature of a constant magnitude of $\partial\phi$ that allows stealth hair, thus we look for an Einstein manifold, $R_{\mu\nu}=\Lambda g_{\mu\nu}$, admitting such a solution for $\phi$. First, the equations of motion under these assumptions become

[TABLE]

for the metric and

[TABLE]

for the scalar, where for compactness we have defined

[TABLE]

Setting aside the problem of finding a solution for $\phi$ momentarily, note that in the above, $A_{3}$, $G$ etc. are all constants, evaluated at some $X=X_{0}=(\nabla\phi)^{2}$. Since $\phi_{\mu}$ itself is not necessarily a constant vector, to satisfy the above in general we must choose subspaces of the general parameter space for $\{A_{3},G,K\}$. Starting with the top line of (4), we deduce $A_{3}(X_{0})=0$ (unless spacetime has very special symmetries as we will see later), hence

[TABLE]

where we emphasise that this is at the specific value of $X$, $X_{0}$. These two constraints now imply that (3) is satisfied, provided we set $\Lambda=-K/(2G)|_{X_{0}}$. In other words, the cosmological constant appearing in the Einstein manifold is no longer the bare cosmological constant included in the constant part of the $K$ function. This is the self-tuning property of these gravity theories.

To sum up: Given a general Lagrangian (1), we first look for zeros of $A_{3}$ that determine the value(s) of $X_{0}$, then ask that the derivatives of $G$ and $K$ are related at $X=X_{0}$ as required above. The effective cosmological constant is then fixed by the ratio of $K$ to $G$.

Having established the conditions under which Einstein-like metrics can be solutions to modified gravity, we now make a key observation that allows us to construct a stealth solution on the Einstein manifold: Given sufficient symmetry in a spacetime, the geodesic equation

[TABLE]

can be solved using a Hamilton-Jacobi potential $S$, such that the gradient of the potential gives the tangent vector of the geodesic

[TABLE]

Typically, this method is used to simplify the solution of a particular geodesic (such as the orbit of a planet), however, the form of the potential can be used over a wider range of co-ordinate values that in the case of a hypersurface orthogonal geodesic congruence becomes effectively the whole of the spacetime. Thus, given that $\phi_{\mu}$ has constant magnitude, as does the tangent vector of an affinely parametrised geodesic, it is natural to make the identification

[TABLE]

the properties of the geodesic congruence then will ensure that $\phi$ has the requisite properties to be a stealth solution to the extended gravity equations of motion. Moreover, this provides a nice physical interpretation of the constants appearing in the solution.

We will now illustrate this technique and find a rotating black hole with stealth hair. Consider the Kerr-(A)dS geometry CARTER1968399

[TABLE]

where

[TABLE]

$M$ is the black hole mass, $a$ the angular momentum parameter and $\ell=\sqrt{3/\Lambda}$ is the de Sitter radius, related to the effective cosmological constant (for AdS reverse the sign of $\ell^{2}$).

Applying the Hamilton-Jacobi technique, we note that the components of the metric (10) are independent of $t$ and $\varphi$, thus $E=-p_{t}$ and $L_{z}=p_{\varphi}$ are two constants of the motion, identified with the energy and the azimuthal angular momentum respectively. A third constant of motion is the magnitude of the tangent vector $g^{\mu\nu}p_{\mu}p_{\nu}=X_{0}=-m^{2}$, associated with the rest mass of the test particle222Note, for illustration we take timelike geodesics, should a spacelike congruence be required, substitute $m^{2}\to-m^{2}$ in the derivation.. Most importantly however, a fourth constant was discovered by Carter Carter:1968rr (here generalised to include $\Lambda$):

[TABLE]

who demonstrated that the geodesic equation was separable. We can therefore write

[TABLE]

where

[TABLE]

with

[TABLE]

Now let us look for explicit solutions for the scalar field $\phi=S$. This places further constraints on the potential, as we require $\phi_{\mu}$ to be regular throughout the spacetime. Checking regularity on the axes requires $\partial S/\partial\theta\to 0$ as $\theta\to 0,\pi$, i.e. $\Theta\propto\text{sin}^{2}\theta$. This in turn requires $L_{z}=0$ and ${\cal Q}+\Xi^{2}a^{2}E^{2}=m^{2}a^{2}$, and writing $\Xi E=\eta m$, we get:

[TABLE]

This has now reduced the parameter space to an overall scaling, $m$, and a “relative energy” $\eta$, constrained to lie in $\eta\in[\eta_{c},1]$; the upper limit coming from $\Theta\geq 0$, and the lower limit from $R\geq 0$ in (17).

At first sight, it appears we have four distinct solutions coming from the choice of signs in (14), however, an interesting restriction occurs when $\eta=1$ or $\eta_{c}$. In this case $\Theta$ (or $R$) vanishes for some value of $\theta$ (or $r$), and the branch choice changes. This is most easily seen for $\eta=1$, here $\Theta_{1}=\frac{m^{2}a^{4}}{\ell^{2}}\text{sin}^{2}\theta\text{cos}^{2}\theta$, and the natural root is $\text{cos}\theta$ which changes sign across the hemisphere. The same phenomenon occurs for $R$, but this leads to an important consequence as we now discuss.

Inspection of (14) shows that $S_{r}\sim\pm m\eta r^{\star}$ near the event horizons, where $r^{\star}=\int dr(r^{2}+a^{2})/\Delta_{r}$ is the tortoise coordinate, therefore, if we interpret $\sqrt{R}$ as being the positive root, our scalar field will be divergent at one or the other horizon (dependent on the branch choice). Note however, that for $\eta_{c}$, $R$ has a quadratic zero at some $r_{0}$: $R\sim R^{\prime\prime}(r_{0})(r-r_{0})^{2}/2$, thus the true root, $\sqrt{R}\sim(r-r_{0})$, changes sign at $r_{0}$. This means that for $\phi$ to be differentiable, we must change the sign of $\sqrt{R}$ across $r_{0}$ and set

[TABLE]

where $H$ is the Heaviside step function. This now renders $\phi$ finite at both future event horizons, and infinitely differentiable between the horizons as shown in figure 1.

It is worth emphasising this last point: All black hole solutions in the literature for higher order scalar-tensor gravity are spherically symmetric, and have scalar fields that diverge either on the black hole or cosmological event horizon. While this is not a physical problem when $\phi$ interacts with gravity only through its gradient, it is nonetheless a less appealing feature of these solutions. Here, we have constructed a rotating black hole with finite stealth scalar hair. This scalar will be manifestly continuous across each horizon, and be straightforward to analyse in perturbation theory. Finally, the integral for the $\theta-$potential $S_{\theta}$ gives

[TABLE]

leading to an “off-centre” behaviour in the scalar as shown in figure 2.

For $\eta>\eta_{c}$, the radial function $R$ has no zero, and the scalar field diverges on one horizon, in common with the known solutions in the literature. The field also demonstrates a similar asymmetry in $\theta$, except for $\eta=1$, when

[TABLE]

rendering the angular variation symmetric about the equator as shown in figure 3.

Our solution for $\phi$ shows a clear dependence on both $\theta$ and $r$, as well as the time dependence in common with known stealth solutions having spherical symmetry Babichev:2013cya . The key difference is that we can construct a scalar that is finite on both the black hole and cosmological horizon. To compare to solutions in the literature, we take spherical symmetry ($a\to 0$) and find general solutions of the form $\phi=-m\eta t\pm S_{r}$. Once again, we have a finite-$\phi$ solution for $\eta_{c}^{2}=1-3(M/\ell)^{2/3}$,

[TABLE]

where $r_{0}=M^{1/3}\ell^{2/3}$. It is interesting to compare this to the time-dependent solution of a black hole in slow-roll inflation Chadburn:2013mta ; Gregory:2018ghc . There, the scalar profile $\phi_{SR}\propto T$ (for a suitable time coordinate) is also finite at both horizons, but (21) has constant gradient, whereas $\phi_{SR}$ solves a wave equation, resulting in a slightly different radial profile between the horizons.

Also, note that in the case of spherical symmetry, we can relax our constraint $A_{3}(X_{0})=0$. In this case ${\cal E}_{3},\,{\cal E}_{4}$ and the Riemann tensor term appearing in (3,4) have a simple form, and combine to require

[TABLE]

with the same self-tuning condition for the cosmological constant333See Babichev:2016kdt for a precise analysis of self tuning conditions in spherical symmetry., however, note that $A_{3}(X_{0})\neq 0$ requires $\eta\equiv 1$, thus we no longer have the $\eta-$degree of freedom. What now happens is that spacetime becomes foliated by surfaces of constant $\phi$ that are flat (for $\Lambda=0$).

Another interesting possibility for spherical symmetry is a static solution, found by setting $E=0$, hence $X_{0}>0$ and $\phi$ corresponds to a congruence of spacelike geodesics:

[TABLE]

agreeing with a solution reported in Motohashi:2019sen .

It is also worth noting a side result of our analysis: A search for solutions with $X=X_{0}$ and spherical symmetry in $c_{T}=1$ theories (1) allows only for Einstein geometries. This is unlike Horndeski theories (with $c_{T}\neq 1$) where solutions of black holes and solitons were found that have $X=X_{0}$ but are not Einstein spaces (for a concise up to date review see Lehebel:2018zga ). This is an interesting feature, hinting that solutions of $c_{T}=1$ theories, where $X$ is not constant belong to branches that eventually flow towards $X$ constant solutions with an Einstein space metric.

To sum up: we have presented the first exact solutions for a rotating black hole with scalar hair in shift-symmetric scalar-tensor theories with unitary speed for gravitational waves. Our method was based on an interesting correspondence between families of black hole geodesics and stealth solutions in modified gravity. The geodesic correspondence can be used to find other stealth solutions, for instance for more general type D spacetimes, or even non-stealth solutions that do not exist in GR – one only needs to compute the Hamilton-Jacobi potential for a geodesic in the relevant spacetime. It is also plausible that this technique can be extended to other modifications of gravity, such as vector-tensor theories where we know that certain stealth solutions exist. Although we did not consider charged black holes, clearly one can also use this method to find stealth Kerr-Newman solutions. Perhaps most importantly, we have presented a scalar solution that is finite at both the cosmological and event horizons, thus manifestly extendible beyond the cosmological horizon. The angular asymmetry of this solution could provide a distinctive signature of this hair, although this would require a full perturbation analysis beyond the scope of this investigation.

Acknowledgments.    It is a pleasure to thank Eugeny Babichev, Gilles Esposito-Farèse, Antoine Lehébel, Karim Noui, Eric Gourgoulhon, Karim Van Aelst for many interesting discussions. CC and NS would like to acknowledge networking support by the GWverse COST Action CA16104, Black holes, gravitational waves and fundamental physics. CC thanks the Laboratory of Astronomy of AUTh in Thessaloniki for hospitality during the course of this work. MC is supported by the Labex P2IO and the Enhanced Eurotalents Fellowship. RG is supported in part by the STFC [consolidated grant ST/P000371/1], and in part by the Perimeter Institute. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research and Innovation.